MTR_500_HW_3 - Try a solution of the form U = xr. What...

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Homework # 2 Due 23rd of March taken (The problems are 1. Reduction 01 onler for double roots. Consider the second order differential equation ay"+by'+cy=O with ~ - 4ac = O. Find one solution, Yl, using the standard technique. Then let 1/2 = V(t)Yl and solve for v(t). Show the resulting general solution for a double root is Y = CIYl + c~tYl In addition, show the two solutions are linearly independent. 2. Solve the following second-order, nonhomogeneous differential equation tI" - 2y' + Y = texp(t) + 4 y(O) = 1, U'(O) = 1 using (a) the method of undetermined coefficients and (b) variation of parameters. 3. Euler equations: consider the equation (x ~ 0) x~y" + oxy' + .8Y = 0 .
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Unformatted text preview: Try a solution of the form U = xr. What values of r satisfy the equation? What is the general solution if the equation for r baa (a) two real, unequal roots, (b) two real, equal roots (use reduction of order), and (c) two, complex conjugate roots (determine the two real solutions). For problems 4-10, (a) determine the eigenvalues and eigenvectors (real solutions), (b) sketch the behavior and classify the behavior. MTR 500 Spring 2005-3-17 Kutz's notes) from Professor J. Nathan )% £'=( £'=( .i'=( .f'=( .f'=( i'=( .i'=( 2 -5 1 -2 4. 5. 6. 7. 8. 9. 10.-1 -1 ) 0 -0.26 :f )x 3 -4 1 -1 2 -5/2 ) 9/5 -1 f 2 -1 ) 3 -2 f 1.;3 ) v'3 -1 i 3 -2 ) 2 -2 f...
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This note was uploaded on 02/14/2010 for the course CHE NGN500 taught by Professor Ghaleb during the Spring '10 term at American Dubai.

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